We are asked to identify the fractions plotted on the number line and to give the answers
as proper or improper fractions, not as mixed numbers.
The first step is to focus on the unit, which is the length from 0 to 1 on the number line.
So this length here is equal to 1 unit and notice how the tick marks cut or divide this
length into 1, 2, 3 equally sized parts or pieces, which means the length between tick
marks is equal to 1/3 of a unit.
Knowing this we can now count to the right of 0 by 1/3's and to the left of 0 by -1/3's.
Counting to the right of 0, we have 0, 1/3, 2/3, 3/3 which is equal to 1, and 4/3.
And now counting to the left of 0, we count by -1/3's.
We have -1/3, -2/3, -3/3 which equals -1, -4/3, -5/3, -6/3 which is equal to -2, -7/3,
and -8/3.
And now we have the information we need to express A, B, and C as proper and improper
fractions.
A equals -7/3.
B is equal to -4/3 and C is equal to positive 2/3.
So these are the answers to the question, but for review let's also express the improper
fractions as mixed numbers.
So looking at -4/3, notice how this is 1/3 of a unit to the left of -1 and therefore
as a mixed number -4/3 is equal to -1 1/3.
And now looking at -7/3, notice how this is 1/3 of a unit to the left of -2.
-7/3 equals -2 1/3.
Again this question does want the fractions as proper fractions or improper fractions
but I think it is worth reviewing the improper fractions as mixed numbers.
Let's look at a second example.
Again the first step is to focus on the length of one unit, which again is the distance from
0 to 1.
Notice how this length is cut or divided into 1, 2, 3, 4 equally sized parts or pieces,
which means the length between tick marks equals 1/4 of a unit.
So now we can count by 1/4's to the right of 0 and we can count by -1/4's to the left
of 0.
Starting and 0 and moving to the right we have 0, 1/4, 2/4 which is equivalent to 1/2,
3/4 and 4/4 which is equivalent to 1.
Now counting to the left of 0 we have 0, -1/4, -2/4 which is equivalent to -1/2, -3/4, -4/4
is equal to -1, -5/4, -6/4 which is equivalent to -3/2, -7/4, -8/4 which is equivalent to
-2 and finally -9/4.
So now we know A is equal to -2.
B is equal to -2/4, which we'll come back to in just a moment because this fraction
does simplify.
And C equals 3/4.
Looking back at the fraction -2/4, more formally this fraction does simplify because 2 and
4 share a common factor of 2.
To simplify we divide the numerator and denominator by 2.
Simplifying we get -1/2.
So -2/4 equals -1/2 so both fractions are correct but the simplified fraction is the
fraction -1/2.
If we look back at the number line and focus on the length from 0 to -1.
We know this length is 1 unit.
If we cut divide this length into 2 equally sized parts or pieces, each of these smaller
lengths is equal to 1/2 of a unit.
So starting at 0 and moving to the left we have 0, -1/2, and then -1.
This shows -2/4 is equivalent to -1/2.
I hope you found this helpful.